Bivariant K-theory of generalized Weyl algebras

We compute the isomorphism class in $\mathfrak{KK}^{alg}$ of all noncommutative generalized Weyl algebras $A=\CC[h](\sigma, P)$, where $\sigma(h)=qh+h_0$ is an automorphism of $\CC[h]$, except when $q\neq 1$ is a root of unity. In particular, we compute the isomorphism class in $\mathfrak{KK}^{alg}$ of the quantum Weyl algebra, the primitive factors $B_{\lambda}$ of $U(\mathfrak{sl}_2)$ and the quantum weighted projective lines $\mathcal{O}(\mathbb{WP}_q(k, l))$.